Answer :
To graph the linear inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex], follow these steps:
### Step 1: Identify the Boundary Line
First, we need to find the boundary line corresponding to the inequality. The boundary line is the equation obtained by changing the inequality sign to an equality:
[tex]\[ \frac{1}{2} x - 2 y = -6 \][/tex]
### Step 2: Solve for [tex]\(y\)[/tex] in Terms of [tex]\(x\)[/tex]
To make it easier to plot, solve the boundary line equation for [tex]\(y\)[/tex]:
[tex]\[ \frac{1}{2} x - 2 y = -6 \][/tex]
Starting from the boundary line equation, we can isolate [tex]\(y\)[/tex]:
[tex]\[ \frac{1}{2} x - 2 y = -6 \implies -2 y = -\frac{1}{2} x - 6 \implies y = \frac{1}{4} x + 3 \][/tex]
Thus, the equation of the boundary line in terms of [tex]\(y\)[/tex] is:
[tex]\[ y = \frac{1}{4} x + 3 \][/tex]
### Step 3: Plot the Boundary Line
Next, plot the boundary line [tex]\(y = \frac{1}{4} x + 3\)[/tex] on a coordinate plane. This is a straight line with a slope of [tex]\(\frac{1}{4}\)[/tex] and a y-intercept of 3.
### Step 4: Determine the Shaded Region
Since the original inequality is [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex], you need to identify the region where this inequality holds.
To determine which side of the boundary line to shade, choose a test point that is not on the line. A common choice is the origin [tex]\((0,0)\)[/tex].
Substitute [tex]\((0,0)\)[/tex] into the original inequality:
[tex]\[ \frac{1}{2} \cdot 0 - 2 \cdot 0 > -6 \implies 0 > -6 \][/tex]
This statement is true, so the region containing the origin is the solution to the inequality.
### Step 5: Shade the Correct Region
Finally, shade the region above the boundary line [tex]\(y = \frac{1}{4} x + 3\)[/tex]. Since the inequality is strict ([tex]\(>\)[/tex] rather than [tex]\(\geq\)[/tex]), you should not include the boundary line itself in the shaded region; thus, use a dashed line to indicate the boundary.
### Summary
The graph of the inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex] includes:
- A dashed boundary line corresponding to [tex]\(y = \frac{1}{4} x + 3\)[/tex]
- The region above this dashed line, which includes the origin [tex]\((0,0)\)[/tex]
### Step 1: Identify the Boundary Line
First, we need to find the boundary line corresponding to the inequality. The boundary line is the equation obtained by changing the inequality sign to an equality:
[tex]\[ \frac{1}{2} x - 2 y = -6 \][/tex]
### Step 2: Solve for [tex]\(y\)[/tex] in Terms of [tex]\(x\)[/tex]
To make it easier to plot, solve the boundary line equation for [tex]\(y\)[/tex]:
[tex]\[ \frac{1}{2} x - 2 y = -6 \][/tex]
Starting from the boundary line equation, we can isolate [tex]\(y\)[/tex]:
[tex]\[ \frac{1}{2} x - 2 y = -6 \implies -2 y = -\frac{1}{2} x - 6 \implies y = \frac{1}{4} x + 3 \][/tex]
Thus, the equation of the boundary line in terms of [tex]\(y\)[/tex] is:
[tex]\[ y = \frac{1}{4} x + 3 \][/tex]
### Step 3: Plot the Boundary Line
Next, plot the boundary line [tex]\(y = \frac{1}{4} x + 3\)[/tex] on a coordinate plane. This is a straight line with a slope of [tex]\(\frac{1}{4}\)[/tex] and a y-intercept of 3.
### Step 4: Determine the Shaded Region
Since the original inequality is [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex], you need to identify the region where this inequality holds.
To determine which side of the boundary line to shade, choose a test point that is not on the line. A common choice is the origin [tex]\((0,0)\)[/tex].
Substitute [tex]\((0,0)\)[/tex] into the original inequality:
[tex]\[ \frac{1}{2} \cdot 0 - 2 \cdot 0 > -6 \implies 0 > -6 \][/tex]
This statement is true, so the region containing the origin is the solution to the inequality.
### Step 5: Shade the Correct Region
Finally, shade the region above the boundary line [tex]\(y = \frac{1}{4} x + 3\)[/tex]. Since the inequality is strict ([tex]\(>\)[/tex] rather than [tex]\(\geq\)[/tex]), you should not include the boundary line itself in the shaded region; thus, use a dashed line to indicate the boundary.
### Summary
The graph of the inequality [tex]\(\frac{1}{2} x - 2 y > -6\)[/tex] includes:
- A dashed boundary line corresponding to [tex]\(y = \frac{1}{4} x + 3\)[/tex]
- The region above this dashed line, which includes the origin [tex]\((0,0)\)[/tex]
